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A024313 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

%I A024313 #15 Jan 09 2024 16:06:28
%S A024313 3,3,10,17,37,59,114,185,334,540,938,1518,2573,4163,6946,11239,18559,
%T A024313 30029,49248,79685,130090,210490,342596,554332,900423,1456915,2363370,
%U A024313 3824013,6197753
%N A024313 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.
%H A024313 G. C. Greubel, <a href="/A024313/b024313.txt">Table of n, a(n) for n = 1..1000</a>
%H A024313 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-1,-1,-3,2,1,1,1).
%F A024313 G.f.: x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F A024313 From _G. C. Greubel_, Jan 17 2022: (Start)
%F A024313 a(2*n) = Lucas(2*n+3) + F(2*n+2) - Lucas(n+3) - (n+1)*F(n+2).
%F A024313 a(2*n+1) = Lucas(2*n+4) + F(2*n+3) - Lucas(n+3) - (n+2)*F(n+2), where F(n) = A000045(n). (End)
%t A024313 a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+3] + F[n+2] - LucasL[n/2 +3] - (n/2 +1)*F[n/2 +2], LucasL[n+3] + F[n+2] - LucasL[(n+5)/2]-(n+3)/2*Fibonacci[(n+3)/2]]];
%t A024313 Table[a[n], {n, 40}] (* _G. C. Greubel_, Jan 17 2022 *)
%o A024313 (Magma)
%o A024313 R<x>:=PowerSeriesRing(Integers(), 40);
%o A024313 Coefficients(R!( x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // _G. C. Greubel_, Jan 17 2022
%o A024313 (Sage)
%o A024313 def A024313_list(prec):
%o A024313     P.<x> = PowerSeriesRing(ZZ, prec)
%o A024313     return P( x*(3 -2*x^2 +4*x^3 -x^4 -3*x^5 -2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
%o A024313 a=A024313_list(41); a[1:] # _G. C. Greubel_, Jan 17 2022
%Y A024313 Cf. A024312, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.
%Y A024313 Cf. A000032, A000045, A023531.
%K A024313 nonn,easy
%O A024313 1,1
%A A024313 _Clark Kimberling_